On Approximability of Linear Ordering and Related NP-optimization Problems on Graphs (Extended Abstract)

We investigate the approximability of minimum and maximum linear ordering problems (MIN-LOP and MAX-LOP) and related feedback set problems such as maximum weight acyclic subdiagraph (MAX-W-SUBDAG), minimum weight feedback arc/vertex set (MIN-W-FAS/ MIN-W-FVS) and a generalization of the latter called MIN-Subset-FAS/MIN-Subset-FVS. MAX-LOP and the other problems have been studied by many researchers but, though MIN-LOP arises in many practical contexts, it appears that it has not been studied before. In this paper, we ;rst note that these minimization (respectively, maximization) problems are equivalent with respect to strict reduction and so have the same approximability properties. While MAX-LOP is APX-complete, MIN-LOP is only APX-hard. We conjecture that MIN-LOP is not in APX, unless P = NP. We obtain a result connecting extreme points of linear ordering polytope with approximability of MIN-LOP via LP-relaxation which seems to suggest that constant factor approximability of MIN-LOP may not be obtainable via this method. We also prove that (a) MIN-Subset-FAS cannot be approximated within a factor (1 − ) log n, for any ¿ 0, unless NP ⊂ DTIME(n log ), and (b) MIN-W-k-FAS, a generalization of MIN-LOP, is NPO-complete. We then show that At-MIN-LOP (respectively, At-MAX-LOP), in which the weight matrix satis;es a parametrized version of a stronger form of triangle inequality, with parameter t ∈ (0; 2], is approximable within a factor (2 + t)=2t (respectively, 4=(2 + t)). We also show that these restricted versions are in PO iD t = 2 and are not in PTAS for t ∈ (0; 2), unless P = NP. ? 2003 Elsevier B.V. All rights reserved.